AP® Calculus AB 2009 Scoring Guidelines Form BThe College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. Founded in 1900, the association is composed of more than 5,600 schools, colleges, universities and other educational organizations. Each year, the College Board serves seven million students and their parents, 23,000 high schools and 3,800 colleges through major programs and services in college readiness, college admissions, guidance, assessment, financial aid, enrollment, and teaching and learning. Among its best-known programs are the SAT®, the PSAT/NMSQT® and the Advanced Placement Program® (AP®). The College Board is committed to the principles of excellence and equity, and that commitment is embodied in all of its programs, services, activities and concerns. © 2009 The College Board. College Board, Advanced Placement Program, AP, AP Central, SAT, and the acorn logo are registered trademarks of the College Board. PSAT/NMSQT is a registered trademark of the College Board and National Merit Scholarship Corporation. Permission to use copyrighted College Board materials may be requested online at: www.collegeboard.com/inquiry/cbpermit.html. Visit the College Board on the Web: www.collegeboard.com. AP Central®is the official online home for AP teachers: apcentral.collegeboard.com. www.mymathscloud.com
AP® CALCULUS AB 2009 SCORING GUIDELINES (Form B) Question 1 © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. At a certain height, a tree trunk has a circular cross section. The radius ()Rt of that cross section grows at a rate modeled by the function ()()213sin centimeters per year16dRtdt=+for 03,t≤≤ where time t is measured in years. At time 0,t= the radius is 6 centimeters. The area of the cross section at time t is denoted by ().At(a) Write an expression, involving an integral, for the radius ()Rt for 03.t≤≤ Use your expression to find ()3.R(b) Find the rate at which the cross-sectional area ()At is increasing at time 3t= years. Indicate units of measure. (c) Evaluate ()30.At dt′∫ Using appropriate units, interpret the meaning of that integral in terms of cross-sectional area. (a) ()()()20163sin16tRtxdx=++⌠⌡()36.610R= or 6.611 3 : ()() 1 : integral 1 : expression for 1 : 3RtR⎧⎪⎨⎪⎩(b) ()()()2AtRtπ=()() ()2AtRtRtπ′′=()238.858 cmyearA′=3 : ()() 1 : expression for 1 : expression for 1 : answer with unitsAtAt⎧⎪′⎨⎪⎩(c) ()( )( )303024.200At dt AA′=−=∫ or 24.201 From time 0t= to 3t= years, the cross-sectional area grows by 24.201 square centimeters. 3 : ()()30301 : uses Fundamental Theorem of Calculus 1 : value of 1 : meaning of At dtAt dt⎧⎪′⎪⎨⎪′⎪⎩∫∫www.mymathscloud.com
AP® CALCULUS AB 2009 SCORING GUIDELINES (Form B) Question 2 © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. A storm washed away sand from a beach, causing the edge of the water to get closer to a nearby road. The rate at which the distance between the road and the edge of the water was changing during the storm is modeled by ()cos3fttt=+ − meters per hour, t hours after the storm began. The edge of the water was 35 meters from the road when the storm began, and the storm lasted 5 hours. The derivative of ()ftis ()1sin .2fttt′=−(a) What was the distance between the road and the edge of the water at the end of the storm? (b) Using correct units, interpret the value ()41.007f′= in terms of the distance between the road and the edge of the water. (c) At what time during the 5 hours of the storm was the distance between the road and the edge of the water decreasing most rapidly? Justify your answer. (d) After the storm, a machine pumped sand back onto the beach so that the distance between the road and the edge of the water was growing at a rate of ()gp meters per day, where p is the number of days since pumping began. Write an equation involving an integral expression whose solution would give the number of days that sand must be pumped to restore the original distance between the road and the edge of the water. (a) ()503526.494ft dt+=∫or 26.495 meters 2 : {1 : integral1 : answer(b) Four hours after the storm began, the rate of change of the distance between the road and the edge of the water is increasing at a rate of 21.007 meters hours .2 : ()1 : interpretation of 4 1 : unitsf′⎧⎨⎩(c) ()0ft′= when 0.66187t= and 2.84038t=The minimum of f for 05t≤≤ may occur at 0, 0.66187, 2.84038, or 5. ()02f=−()0.661871.39760f=−()2.840382.26963f=−()50.48027f=−The distance between the road and the edge of the water was decreasing most rapidly at time 2.840t=hours after the storm began. 3 : ()1 : considers 0 1 : answer 1 : justificationft′=⎧⎪⎨⎪⎩(d) ()( )500xftdtgpdp−=∫∫2 : {1 : integral of 1 : answergwww.mymathscloud.com
